Off-Shell Quantum Mechanics as Factorization Algebras on Intervals
Dec 9, 2024Citations per year
Abstract: (arXiv)
We present, for the harmonic oscillator and the spin- system, an alternative formulation of quantum mechanics that is `off-shell': it is based on classical off-shell configurations and thus similar to the path integral. The core elements are Batalin-Vilkovisky (BV) algebras and factorization algebras, following a program by Costello and Gwilliam. The BV algebras are the spaces of quantum observables given by the symmetric algebra of polynomials in compactly supported functions on some interval , which can be viewed as functionals on the dynamical variables. Generalizing associative algebras, factorization algebras include in their data a topological space, which here is , and an assignment of a vector space to each open set, which here is the assignment of to each open interval . The central structure maps are bilinear for disjoint intervals and contained in an interval , which here is the wedge product of the symmetric algebra. We prove, as the central result of this paper, that this factorization algebra is quasi-isomorphic to the factorization algebra of `on-shell' quantum mechanics. In this we extend previous work by including half-open and closed intervals, and by generalizing to the spin- system.Note:
- 79 pages, 2 figures
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Figures(3)
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